Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/251

that if $u'$ or $v'$ is discontinuous at the interface, the value of $u''$ or $v''$ becomes in a sense infinite, i.e., ${\text{curl curl }}{\mathfrak {E}}$ and therefore by (6) $\Psi {\ddot {\mathfrak {E}}},$ will be infinite. Now both ${\mathfrak {E}}$ and $\Psi$ are discontinuous at the interface, but infinite values for $\Psi {\ddot {\mathfrak {E}}}$ are not admissible. Therefore $u'$ and $v'$ are continuous. Again, if $u$ or $v$ is discontinuous, $u'$ or $v'$ will become infinite, and therefore $u''$ or $v''.$ Therefore $u$ and $v$ are continuous These conditions may be expressed in the most general manner by saying that the components of ${\mathfrak {E}}$ and ${\text{curl }}{\mathfrak {E}}$ parallel to the interface are continuous. This gives four complex scalar conditions, or in all eight scalar conditions, for the motion at the interface, which are sufficient to determine the amplitude and phase of the two reflected and the two refracted rays in the most general case. It is easy, however, to deduce from these four complex conditions, two others, which are interesting and sometimes convenient. It is evident from the definitions of $w'$ and $w''$ that if $w,v,u'$ and $v'$ are continuous at the interface $w'$ and $w''$ will also be continuous. Now $-w''$ is equal to the component of $\Psi {\ddot {\mathfrak {E}}}$ normal to the interface. The following quantities are therefore continuous at the interface:

the components parallel to the interface of	${\mathfrak {E}},$	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \ \end{matrix}}\right\}\,}$ (7)
the component normal to the interface of	$\Psi {\mathfrak {E}},$
all components of curl	${\text{curl }}{\mathfrak {E}}.$

To compare these results with those derived from the electrical theory, we may take the general equation of monochromatic light on the electrical hypothesis from a paper in a former volume of this Journal. This equation, which with an unessential difference of notation may be written^[1]

$-{\text{Pot }}{\ddot {\mathfrak {F}}}-\nabla {\text{Q}}=4\pi \Phi {\mathfrak {F}},$

(8)

was established by a method and considerations similar to those which have been used to establish equation (6), except that the ordinary law of electrodynamic induction had the place of the new law of elasticity. ${\mathfrak {F}}$ is a complex vector representing the electrical displacement as a harmonic function of the time; $\Phi$ is a complex linear vector operator, such that $4\pi \Phi {\mathfrak {F}}$ represents the electromotive force necessary to keep up the vibration ${\mathfrak {F}}.\,{\text{Q}}$ is a complex scalar representing the electrostatic potential, $\nabla {\text{Q}}$ the vector of which the three components are

${\frac {d{\text{Q}}}{dx}},$⁠${\frac {d{\text{Q}}}{dy}},$⁠${\frac {d{\text{Q}}}{dz}}\cdot$

Pot denotes the operation by which in the theory of gravitation the potential is calculated from the density of matter.^[2] When it is

1. See page 218 of this volume, equation (12).
2. The symbol $-{\text{Pot}}$ is therefore equivalent to $4\pi \nabla ^{-2}$ as used by Sir William Thomson (with a happy economy of symbols) at the last meeting of British Association to express the same law of eleotrodynamio induction, except that the symbol is here used as a vector operator. See Nature, vol. xxxviii, p. 571, sub init.